Example 1

For the integral, (a) identify u and u' and (b) integrate by substitution.

Answer

(a) The most complicated "inside" function is 6x + 4, so let

u = 6x + 4u' = 6

(b) First we change variables. We have to break up 30 into 5 ⋅ 6 so we can find u':

We use an appropriate pattern to integrate:

And finally put the original variable back in:

(u)5 + C = (6x + 4)5 + C

We conclude

Example 2

For the integral, (a) identify u and u' and (b) integrate by substitution.

Answer

(a) The exponent of 4 is the most complicated inside function:

so we have

u = x2 + 4u' = 2x

(b) We change variables, integrate, and put the original variable back in:

Example 3

For the integral, (a) identify u and u' and (b) integrate by substitution.

Answer

(a) The most complicated "inside" function is the ex inside the sin function:

The derivative of ex is ex, so

u = exu' = ex

(b) Change variables, integrate, and change variables back:

Example 4

Integrate. The problem may or may not require substitution.

Answer

The most complicated function that's still inside something is the denominator of the fraction, which is "inside" the operation :

Let u = x + 1. Then u' = 1. We have

Example 5

Integrate. The problem may or may not require substitution.

Answer

The most complicated inside function is the exponent of e:

Let u = sin x. Then u' = cos x.

Example 6

Integrate. The problem may or may not require substitution.

Answer

This integral doesn't require substitution. It only requires thinking backwards. What has a derivative of -cos x? The answer is -sin x. So

Example 7

Integrate. The problem may or may not require substitution.

Answer

We can rewrite this integral as

This makes it more obvious that we should choose u = sin x and u' = cos x.

Example 8

Integrate. The problem may or may not require substitution.

Answer

The most complicated inside function is the exponent of 3, so let u = (2x2 + 4x) and then u' = (4x + x).

Example 9

Integrate. The problem may or may not require substitution.

Hint

Factor 20.

Answer

Let u = (2x + 3). Then u' should be 2. In order to see 2 as a factor of the integrand, we have to break up the coefficient 20:

Now we have to think backwards. What has a derivative of 10u4? Answer: 2u5. So

Example 10

Integrate. The problem may or may not require substitution.

Hint

Simplify the integrand.

Answer

This doesn't require substitution. Simplify the integrand by raising 5 and x to the 4th power:

What has a derivative of 625x4? Answer:

So

Example 11

Integrate. The problem may or may not require substitution.

Answer

If we only looked at the first guideline for how to choose u, we would take cos (lnx) as u since that's the most complicated thing that's still "inside" something else (it's the numerator of the fraction). However, the derivative of cos (lnx) is not a factor of the integrand, so that won't work. Let's try making u a little less complicated. Let

Then

Example 12

Integrate. The problem may or may not require substitution.

Answer

Following the hint, rewrite the integral:

We have two choices. If we let u = sin x and u' = cos x, we get the integral

which isn't very helpful. On the other hand, if we let u = cos x, then u' = -sin x and we get an integral we know what to do with:

Example 13

Integrate. The problem may or may not require substitution.

Answer

The most obvious choice of "inside" function is u = (3x2 + 4x + 7).

Then u' = (6x + 4).

Example 14

Integrate.

Answer

Let

u = 3x + 4

u' = 3

We multiply by 3 inside the integral and without. Since we're not changing the value of the expression:

We integrate, writing C instead of , and put the original variable back in:

Example 15

Integrate.

Answer

Let

In this case we need the factor inside the integral, and its reciprocal 4 outside the integral.

Example 16

Integrate.

Answer

Let

u = e4x

u' = 4e4x

We need to introduce the factor 4 to the integrand, so we multiply the integrand by 4 and the outside of the integral by .

Those parentheses in the second-to-last step are important. If we had written

instead of

we would have gotten the wrong answer!

Example 17

Integrate.

Answer

Let u = (3x2 + 4) and u' = 6x. Then we need to introduce 6 to the integrand and to the outside of the integral.

Example 18

Integrate.

Hint

Factor .

Answer

Let u = (7x – 12) so u' = 7. We do have a factor of 7 in the integrand, which we can see if we factor 35.

Example 19

Integrate.

Answer

Let . Then . We factor so we can see u'.

The other factor of isn't helpful, so let's move it outside of the integral:

The simplest antiderivative of u6⋅ u' is , so

Again we have and .

This time, we do want to move the factor 7 into the integrand since it's easy to find an antiderivative of 7u6u'.

Example 20

Integrate.

Hint

(-1)(-1) = 1

Answer

Let . Since , we have

We rewrite the integrand:

This is almost what we want, but the sign is wrong. Since (-1)(-1) = 1, let's multiply the integrand by (-1)(-1). Then we can see u' in the integrand:

Example 21

Integrate.

Answer

Let u = x2 + 4x and u' = 2x + 4. If we multiply the numerator of the integrand by 2, we get u'.

Example 22

Integrate.

Hint

Answer

Following the hint, let's rewrite the integrand:

We choose u to be the denominator, because we know how to find the integral of but we don't know how to find the integral of . So let